Showing papers on "Ladder operator published in 1974"
255 citations
TL;DR: In this article, a modified SCF theory is developed, where by choosing an appropriate operator one can get any desired modified orbitals and their associated orbital energies under orbital transformations, and the effective Hamiltonian is derived.
Abstract: The coupling operator method in the general SCF theory is discussed in terms of the projection operator property of the density operator. The advantage of using the density operator is that one can put the arbitrariness of the general SCF orbitals in evidence. We showed how to put the fundamental condition for the optimum orbitals of the general SCF theory into a more general and useful form. The essential point of the coupling operator is how the variational conditions are included as the projections onto the intermanifolds. By using the arbitrariness of the manifold, we indicate how a modified SCF theory is developed, where by choosing an appropriate operator one can get any desired modified orbitals and their associated orbital energies under orbital transformations. The Appendix contains an extension of Koopmans' theorem as an application of the modified SCF theory. Finally the effective Hamiltonian is derived which is valid for almost all of the proposed SCF theory.
76 citations
TL;DR: In this article, a master equation for the density operator of the light fields alone is derived and analytic solutions are obtained for the diagonal matrix elements of density operator in the Fock representation.
Abstract: The nonlinear interaction of light with matter is described from a quantum-statistical point of view. The phenomena of two-photon emission and two-photon absorption including both the single- and two-mode cases and the Raman effect are discussed in detail. A master equation for the density operator of the light fields alone is derived. This operator equation is converted to a c number equation and analytic solutions are obtained for the diagonal matrix elements of the density operator in the Fock representation. No linearizing approximation is introduced. These solutions allow one to compute the moments of the photon distribution for the above nonlinear processes.
72 citations
41 citations
TL;DR: In this paper, it was shown that a closed form expression of any matrix element on the basis of the eigenfunctions of any factorizable equation can be easily derived from the calculation of one unique particular integral.
Abstract: Within the Schrodinger‐Infeld‐Hull factorization scheme, it is shown that, by suitable transformations, the ``accelerated'' or ``ν‐step'' ladder operator can always be brought to a simple canonical form, i.e., the νth derivative operation. Thus, one obtains a closed form expression of the eigenfunctions involving a Rodrigues' formula. The necessary and sufficient condition that this Rodrigues' formula generates classical orthogonal polynomials is found to be equivalent to the factorizability condition. Consequently, a closed form expression of any matrix element (diagonal or off‐diagonal) on the basis of the eigenfunctions of any factorizable equation is easily derived from the calculation of one unique particular integral. In most cases, this last integral is known analytically. The Kepler problem is reinvestigated as an example. As a concluding remark, further applications of the method are considered.
38 citations
TL;DR: In this article, the inverse of a linear operator on a vector is given by a recursion method which involves only three vectors at a time and is independent of any inner product.
Abstract: The result of the inverse of a linear operator on a vector is given by a recursion method which involves only three vectors at a time. The method is independent of any inner product.
32 citations
Journal Article•
22 citations
TL;DR: In this paper, a group theoretic method based on the results of Winternitz et al. was developed to compute and classify all first and second-order raising and lowering operators admitted by Hamiltonians of the form H¯=−(1/2)Δ2+V(x,y).
Abstract: We develop a group theoretic method based on results of Winternitz et al. to compute and classify all first‐ and second‐order raising and lowering operators admitted by Hamiltonians of the form H¯=−(1/2)Δ2+V(x,y). The key to our results, which generalize to higher dimensions, is a proof that H¯ admits a second‐order raising operator only if the Schrodinger equation separates in Cartesian, polar, or elliptic coordinates.
21 citations
01 Jan 1974
21 citations
TL;DR: In this article, it was shown that the Hamiltonian of the quantal oscillator is equivalent to half an harmonic oscillator for the odd and even eigenspaces separately, and that it can be used to study the correspondent N body problem.
Abstract: In the present work we study the differential operator H=−(1/2)d2/dx2 +m2x2/2+g/x2. This operator known as the Hamiltonian of the quantal oscillator has been a matter of study since the beginning of quantum mechanics. Recently, it has become again actual after the paper of Calogero where the correspondent N body problem (developed in many works) is studied. Parisi and the author have used H as Hamiltonian, studying the anomalous dimensions in one‐dimensional quantum field theory. Finally, Klauder, using H as a simple degree of freedom example, has studied some qualitative features of quantum theories with singular interaction potentials. In the following work we are going to study H, showing that H is equivalent to ``half an harmonic oscillator'' for the odd and even eigenspaces separately.
TL;DR: In this paper, two exactly soluble models, Lipkin's model and a pairing force model, are studied to test a previously introduced particle-hole method (the hermitian operator method) for the description of collective states.
TL;DR: In this paper, a second-order elliptic differential operator coinciding with the Laplace operator in a neighborhood of infinity is considered and an asymptotic approximation with respect to smoothness to the function is constructed by Hadamard's method.
Abstract: Let , , be a second-order elliptic differential operator coinciding with the Laplace operator in a neighborhood of infinity. Let be the Green's function of the Cauchy problem for the operator . Under certain assumptions regarding the trajectories of the Hamiltonian system connected with the operator in question, the following results are obtained: 1) an asymptotic approximation with respect to smoothness to the function is constructed by Hadamard's method, 2) we show that the Fourier transformation of from to is an analytic function of in the complex plane with a cut along the negative part of the imaginary axis, and with and it gives the asymptotic behavior of the fundamental solution of the operator , 3) the asymptotic behavior as of the solutions of the nonstationary problem is obtained.Bibliography: 44 titles.
TL;DR: In this article, it was shown that the transitivity of a strictly cyclic algebra implies its strict (and hence even its strong) density, and that the invariant subspaces of strictlycyclic operator algebras are invariant.
Abstract: We obtain results concerning the invariant subspaces of strictly cyclic operator algebras. In particular, we show that transitivity of a strictly cyclic algebra implies its strict (and hence even its strong) density.
TL;DR: In this article, a self-adjoint pseudo-differential operator P =p(x, D) is constructed from its principal symbol, and the following properties are established: (i) F+F~=Id.
Abstract: Among many problems concerning pseudo-differential operators, one of the most interesting problem is \"to what extent does the symbol function p(x, ξ) describe the spectral properties of an operator p(x> D) ?\" Motivation of this paper comes from this problem. Actually what we do in this note is the following: Assume that P=p(x, D) is a self-adjoint pseudo-differential operator of class L? 0 of Hϋrmander [4]. Then starting from its principal symbol, we explicitly construct self-adjoint operators P + , P~~, R> F and F~ with the following properties; ( i) F+F~=Id. (ii) P=P-P-+R. (iii) P + , P~ and F+y F~ are non-negative self-adjoint operators, (iv) We have the following estimates;
TL;DR: In this paper, the authors obtained the details of the unitary operator and applied it to find its implications on the mass spectrum at low order in θ, and found that the (mass) 2 equation 3 m B 2 = 2 m A 2 2 + m A 0 2 holds up to corrections of order θ 4.
TL;DR: A number operator for a Weyl system is called renormalized essentially if it is obtained from the total number operator by subtraction of a (possibly infinite) constant (in exponentiated form) as discussed by the authors.
Abstract: A number operator for a Weyl system is called renormalized essentially if it is obtained from the total number operator by subtraction of a (possibly infinite) constant (in exponentiated form). Necessary and sufficient conditions for the existence of a renormalized number operator are obtained.